Theorem al0ssb 5309

Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
al0ssb	⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Distinct variable group:   𝑦,𝑋

Proof of Theorem al0ssb

Step	Hyp	Ref	Expression
1		0ex 5308	. . 3 ⊢ ∅ ∈ V
2		sseq2 4009	. . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅))
3		ss0b 4398	. . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅)
4	2, 3	bitrdi 287	. . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅))
5	1, 4	spcv 3596	. 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅)
6		0ss 4397	. . . 4 ⊢ ∅ ⊆ 𝑦
7	6	ax-gen 1798	. . 3 ⊢ ∀𝑦∅ ⊆ 𝑦
8		sseq1 4008	. . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
9	8	albidv 1924	. . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦))
10	7, 9	mpbiri 258	. 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦)
11	5, 10	impbii 208	1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Syntax hints:   ↔ wb 205  ∀wal 1540   = wceq 1542   ⊆ wss 3949  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324

This theorem is referenced by:  iota0def  45796  aiota0def  45852